Complexity and rigidity of Ulrich modules, and some applications

TL;DR

This paper investigates the properties of Ulrich modules over Cohen-Macaulay local rings, revealing their maximal complexity and curvature, and explores their applications as test modules for detecting finite homological dimensions.

Contribution

It provides new characterizations of local rings using Ulrich modules and establishes their Tor-rigidity properties, extending understanding of their homological behavior.

Findings

Ulrich modules over CM local rings have maximal complexity and curvature.

Every Ulrich module of dimension s is (s+1)-Tor-rigid-test but not s-Tor-rigid.

Studied Tor rigidity over deformations of CM local rings of minimal multiplicity.

Abstract

We analyze whether Ulrich modules, not necessarily maximal CM (Cohen-Macaulay), can be used as test modules, which detect finite homological dimensions of modules. We prove that Ulrich modules over CM local rings have maximal complexity and curvature. Various new characterizations of local rings are provided in terms of Ulrich modules. We show that every Ulrich module of dimension $s$ over a local ring is $(s+1)$-Tor-rigid-test, but not $s$-Tor-rigid in general (where $s\geq 1$). Over a deformation of a CM local ring of minimal multiplicity, we also study Tor rigidity.